Superconformal Weight Shifting Operators

TL;DR

This paper introduces a superconformal weight shifting operator framework using analytic superspace, enabling the derivation of superconformal blocks for various correlators in 4D supersymmetric theories.

Contribution

It generalizes conformal weight-shifting operators to superconformal cases, facilitating the construction of superconformal blocks from known half-BPS blocks.

Findings

Constructed SU(m,m|2n)-covariant differential operators.

Derived superconformal blocks from half-BPS blocks.

Presented a superspace formalism as an alternative to embedding space.

Abstract

We develop a framework for constructing superconformal blocks for correlators of general supermultiplets in theories with $\mathrm{SU}(m,m|2n)$ symmetry, such as four-dimensional $\mathcal{N}=2$ and $\mathcal{N} = 4$ conformal theories. We use analytic superspace, viewed as the super-Grassmannian $\mathrm{Gr}(m|n,2m|2n)$, which includes 4D Minkowski space ($m=2,n=0$). In this formalism, superblocks for non-half-BPS correlators are analogous to non-supersymmetric conformal blocks for correlators of fields with spin. We construct $\mathrm{SU}(m,m|2n)$-covariant differential operators which generalise the existing conformal weight-shifting operators, and thus allow us to derive all superconformal blocks from the known half-BPS blocks. Our results provide a framework from which to advance the conformal bootstrap in 4D supersymmetric settings, with potential extensions to lower and higher-dimensional SCFTs. The Grassmannian formalism is also seen to offer a natural and often simpler alternative to the embedding space formalism of non-supersymmetric CFTs.